What is the derivative of this function #y=sec^-1(x^7)#?

1 Answer
Dec 2, 2016

# dy/dx = 7/(xsqrt(x^14-1)) #

Explanation:

When tackling the derivative of inverse trig functions. I prefer to rearrange and use Implicit differentiation as I always get the inverse derivatives muddled up, and this way I do not need to remember the inverse derivatives. If you can remember the inverse derivatives then you use the chain rule.

Let #y=sec^-1(x^7) <=> secy=x^7 #

Differentiate Implicitly:

# secytanydy/dx = 7x^6 #
# :. x^7tanydy/dx = 7x^6 #
# :. tanydy/dx = 7/x # # (x!=0)# .... [1]

Using the #sec"/"tan# identity;

# tan^2y+1=sec^2y #
# :. tan^2y+1=(x^7)^2 #
# :. tan^2y=x^14-1 #
# :. tany=sqrt(x^14-1) #

Substituting into [1]
# :. sqrt(x^14-1)dy/dx = 7/x #
# :. dy/dx = 7/(xsqrt(x^14-1)) #